A082030 -id:A082030 - cp's OEIS frontend

A001340 E.g.f.: 2*exp(x)/(1-x)^3.

2, 8, 38, 212, 1370, 10112, 84158, 780908, 8000882, 89763320, 1094915222, 14431179908, 204423631178, 3097603939952, 50001759773870, 856665220770332, 15526612798028258, 296825612428239848, 5969385443426556422, 125983618731675924020, 2784204907403441680442

Offset: 0

N. J. A. Sloane and Simon Plouffe

a(n) = A001339 (n+1) - A001339 (n)..3-1=2, 11-3=8, 49-11=38... [Gary Detlefs, Jun 06 2010]

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
E. Biondi, L. Divieti, G. Guardabassi, Counting paths, circuits, chains and cycles in graphs: A unified approach, Canad. J. Math. 22 1970 22-35.
Philip Feinsilver and John McSorley, Zeons, Permanents, the Johnson Scheme, and Generalized Derangements, International Journal of Combinatorics, Volume 2011, Article ID 539030, 29 pages; doi:10.1155/2011/539030.

nn = 20; Range[0, nn]! CoefficientList[Series[2*Exp[x]/(1 - x)^3, {x, 0, nn}], x] (* T. D. Noe, Jun 28 2012 *)

a(n) = 2 * A082030(n).
a(n) = floor((n+1)*(n+1)!*e) - floor(n*n!*e) [Gary Detlefs, Jun 06 2010]
a(n) = {exp(1)*(n^2+n+1)*n!} for n>0, where {x} is the neareast integer, proposed by Simon Plouffe, March 1993.
G.f.: (1-x)/x/Q(0) -1/x, where Q(k)= 1 - x - x*(k+2)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013
G.f.: W(0)/x - 1/x, where W(k) = 1 - x*(k+2)/( x*(k+3) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013
Conjecture: a(n) +(-n-3)*a(n-1) +(n-1)*a(n-2)=0. - R. J. Mathar, May 03 2017

Error in description corrected Jan 30 2008

More terms from N. J. A. Sloane, Jan 30 2008

A377811 E.g.f. satisfies A(x) = exp(x * A(x)) / (1 - x)^3.

Original entry on oeis.org

1, 4, 27, 283, 4217, 82971, 2041855, 60475885, 2096566449, 83324680435, 3736041351311, 186594364199277, 10274269171279657, 618386703880855339, 40393224245061185919, 2846030947359659421901, 215160957844217080056161, 17373449685399138641312739, 1492298627191467511376377999

Offset: 0

Seiichi Manyama, Nov 08 2024

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A352410, A377810.

Cf. A082030, A367789, A377743.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x)^3))/(1-x)^3))

a(n) = n!*sum(k=0, n, (k+1)^(k-1)*binomial(n+2*k+2, n-k)/k!);

E.g.f.: exp( -LambertW(-x/(1-x)^3) )/(1-x)^3.
E.g.f.: -LambertW(-x/(1-x)^3)/x.
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(n+2*k+2,n-k)/k!.

A095722 E.g.f.: exp(x)/(1-x)^8.

Original entry on oeis.org

1, 9, 89, 961, 11265, 142601, 1940089, 28245729, 438351041, 7226001865, 126122874201, 2324074591169, 45094140207169, 919088049256521, 19633713260950265, 438708172312264801, 10234490436580101249

Offset: 0

Philippe Deléham, Jul 08 2004

nonn

Sum_{k = 0..n} A094816(n,k)*x^k gives A000522(n), A001339(n), A082030(n), A095000(n), A095177(n), A096307(n), A096341(n) for x = 1, 2, 3, 4, 5, 6, 7 respectively.

G. C. Greubel, Table of n, a(n) for n = 0..250

With[{nn=20},CoefficientList[Series[Exp[x]/(1-x)^8,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, May 26 2013 *)
Table[HypergeometricPFQ[{8, -n}, {}, -1], {n, 0, 20}] (* Benedict W. J. Irwin, May 27 2016 *)

a(n) = Sum_{k = 0..n} A094816(n, k)*8^k.
a(n) = Sum_{k = 0..n} binomial(n, k)*(k+7)! / 7!.
a(n) = 2F0(8,-n;;-1). - Benedict W. J. Irwin, May 27 2016

A082031 Expansion of e.g.f. exp(2*x)/(1-x)^3.

Original entry on oeis.org

1, 5, 28, 176, 1240, 9752, 85120, 819296, 8639872, 99209600, 1233416704, 16517058560, 237137769472, 3634932675584, 59263206154240, 1024222802014208, 18706559855656960, 360062627304341504, 7285354765603176448

Offset: 0

Paul Barry, Apr 02 2003

Binomial transform of A082030

Cf. A052124, A082030.

With[{nn=20},CoefficientList[Series[Exp[2x]/(1-x)^3,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 28 2013 *)

E.g.f.: exp(2*x)/(1-x)^3.
Conjecture: a(n) +(-n-4)*a(n-1) +2*(n-1)*a(n-2)=0. - R. J. Mathar, Nov 24 2012
From Peter Bala, Sep 20 2013: (Start)
a(n) = (1/2)*( Sum_{k = 0..n} (k+2)!*binomial(n,k)*2^(n-k) ).
Based on this series the ZeilbergerRecurrence command in Maple 17 produces the first-order recurrence (n^2 - 3*n + 4)*a(n) = 2^(n+2) + n*(n^2 - n + 2)*a(n-1). Using this it is easy to verify the second-order recurrence conjectured above by Mathar.
The sequence b(n) := n!*(1 + n*(n-1)/2) = n!*A000124(n-1) also satisfies Mathar's recurrence equation but with starting values b(0) = b(1) = 1. This yields the finite continued fraction expansion a(n)/b(n) = 1/(1 - 4/(5 - 2/(6 - 4/(7 - ... - (2*n - 2)/(n + 4) )))), valid for n >= 2.
Lim_{n -> infinity} a(n)/b(n) = e^2 = 1/(1 - 4/(5 - 2/(6 - 4/(7 - ... - (2*n - 2)/(n + 4 - ...))))).
It can be shown that a(n+1)/b(n+1) = 1 + 16*( Sum_{k = 0..n} 2^k/((k + 1)!*(k^4 + 3*k^2 + 4)) ). Taking the limit gives the series acceleration result e^2 = 1 + 16*( Sum_{k = 0..infinity} 2^k/((k+1)!*(k^4 + 3*k^2 + 4)) ). Cf. A082030 and A052124. (End)

A095740 E.g.f.: exp(x)/(1-x)^9.

Original entry on oeis.org

1, 10, 109, 1288, 16417, 224686, 3288205, 51263164, 848456353, 14862109042, 274743964621, 5346258202000, 109249238631169, 2339328151461718, 52384307381414317, 1224472783033479556, 29826054965115774145

Offset: 0

Philippe Deléham Jul 09 2004

nonn

Sum_{k = 0..n} A094816(n,k)*x^k gives A000522(n), A001339(n), A082030(n), A095000(n), A095177(n), A096307(n), A096341(n), A095722(n) for x = 1, 2, 3, 4, 5, 6, 7, 8.

Robert Israel, Table of n, a(n) for n = 0..440

seq(simplify(hypergeom([9,-n],[],-1)),n=0..30); # Robert Israel, May 27 2016

Table[HypergeometricPFQ[{9, -n}, {}, -1], {n, 0, 20}] (* Benedict W. J. Irwin, May 27 2016 *)

a(n) = Sum_{k = 0..n} A094816(n, k)*9^k.
a(n) = Sum_{k = 0..n} binomial(n, k)*(k+8)!/8!.
a(n) = 2F0(9,-n;;-1). - Benedict W. J. Irwin, May 27 2016
a(n) = ((n^8 + 28*n^7 + 350*n^6 + 2492*n^5 + 10899*n^4 + 29596*n^3 + 48082*n^2 + 42048*n + 14833) * Gamma(n+1,1)*e + n^7 + 28*n^6 + 349*n^5 + 2465*n^4 + 10579*n^3 + 27501*n^2 + 40132*n + 25487) / 40320. - Robert Israel, May 27 2016

A367962 Triangle read by rows. T(n, k) = Sum_{j=0..k} (n!/j!).

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 6, 12, 15, 16, 24, 48, 60, 64, 65, 120, 240, 300, 320, 325, 326, 720, 1440, 1800, 1920, 1950, 1956, 1957, 5040, 10080, 12600, 13440, 13650, 13692, 13699, 13700, 40320, 80640, 100800, 107520, 109200, 109536, 109592, 109600, 109601

Offset: 0

Peter Luschny, Dec 06 2023

			  [0]   1;
  [1]   1,    2;
  [2]   2,    4,    5;
  [3]   6,   12,   15,   16;
  [4]  24,   48,   60,   64,   65;
  [5] 120,  240,  300,  320,  325,  326;
  [6] 720, 1440, 1800, 1920, 1950, 1956, 1957;

Cf. A094587, A000142 (T(n, 0)), A052849 (T(n, 1)), A000522 (T(n, n)), A007526 (T(n,n-1)), A038154 (T(n, n-2)), A355268 (T(n, n/2)), A367963(n) (T(2*n, n)/n!).

Cf. A001339 (row sums), A087208 (alternating row sums), A082030 (accumulated sums), A053482, A331689.

T := (n, k) -> add(n!/j!, j = 0..k):
seq(seq(T(n, k), k = 0..n), n = 0..9);

Module[{n=1},NestList[Append[n#,1+Last[#]n++]&,{1},10]] (* or *)
Table[Sum[n!/j!,{j,0,k}],{n,0,10},{k,0,n}] (* Paolo Xausa, Dec 07 2023 *)

from functools import cache
@cache
def a_row(n: int) -> list[int]:
    if n == 0: return [1]
    row = a_row(n - 1) + [0]
    for k in range(n): row[k] *= n
    row[n] = row[n - 1] + 1
    return row

def T(n, k): return sum(falling_factorial(n, n - j) for j in range(k + 1))
for n in range(9): print([T(n, k) for k in range(n + 1)])

T(n, k) = A094587(n, k) * A000522(k).
T(n, k) = e * (n! / k!) * Gamma(k + 1, 1).
Sum_{k=0..n} T(n, k) * 2^(n - k) = A053482(n).
Sum_{k=0..n} T(n, k) * binomial(n, k) = A331689(n).
Recurrence: T(n, n) = T(n, n-1) + 1 starting with T(0, 0) = 1.
For k <> n: T(n, k) = n * T(n-1, k).

A267849 Triangular array: T(n,k) is the 2-row creation rook number to place k rooks on a 3 x n board.

Original entry on oeis.org

1, 1, 3, 1, 6, 12, 1, 9, 36, 60, 1, 12, 72, 240, 360, 1, 15, 120, 600, 1800, 2520, 1, 18, 180, 1200, 5400, 15120, 20160, 1, 21, 252, 2100, 12600, 52920, 141120, 181440, 1, 24, 336, 3360, 25200, 141120, 564480, 1451520, 1814400, 1, 27, 432, 5040, 45360, 317520, 1693440, 6531840, 16329600, 19958400

Offset: 0

Ken Joffaniel M. Gonzales, Jan 21 2016

T(n,k) is the number of ways to place k rooks in a 3 x n Ferrers board (or diagram) under the Goldman-Haglund i-row creation rook mode for i=2. All row heights are 3.

			The triangle T(n,k) begins in row n=0 with columns 0<=k<=n:
     1
     1      3
     1      6     12
     1      9     36     60
     1     12     72    240    360
     1     15    120    600   1800   2520
     1     18    180   1200   5400  15120  20160
     1     21    252   2100  12600  52920 141120 181440
     1     24    336   3360  25200 141120 564480 1451520 1814400
     1     27    432   5040  45360 317520 1693440 6531840 16329600 19958400

Jay Goldman and James Haglund, Generalized rook polynomials, J. Combin. Theory A 91 (2000), 509-530.

Cf. A013610 (1-rook coefficients on the 3xn board), A121757 (2-rook coeffs. on the 2xn board), A013609 (1-rook coeffs. on the 2xn board), A013611 (1-rook coeffs. on the 4xn board), A008279 (2-rook coeffs. on the 1xn board), A082030 (row sums?), A049598 (column k=2), A007531 (column k=3 w/o factor 10), A001710 (diagonal?).

T(n,k) = T(n-1,k) + (k+2) T(n-1,k-1) subject to T(0,0)=1, T(n,k)=0 for n

Triangle simplified (reversing rows, offset 0). - R. J. Mathar, May 03 2017

Previous Showing 11-17 of 17 results.

cp's OEIS Frontend

A001340 E.g.f.: 2*exp(x)/(1-x)^3.

Views

Author

Keywords

Comments

References

Links

Programs

Mathematica

Formula

Extensions

A377811 E.g.f. satisfies A(x) = exp(x * A(x)) / (1 - x)^3.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A095722 E.g.f.: exp(x)/(1-x)^8.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Formula

A082031 Expansion of e.g.f. exp(2*x)/(1-x)^3.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A095740 E.g.f.: exp(x)/(1-x)^9.

Views

Author

Keywords

Comments

Links

Programs

Maple

Mathematica

Formula

A367962 Triangle read by rows. T(n, k) = Sum_{j=0..k} (n!/j!).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Python

SageMath

Formula

A267849 Triangular array: T(n,k) is the 2-row creation rook number to place k rooks on a 3 x n board.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions